Digital Library of Mathematical Functions
About the Project
NIST
25 Zeta and Related FunctionsRiemann Zeta Function

§25.10 Zeros

Contents

§25.10(i) Distribution

The product representation (25.2.11) implies \mathop{\zeta\/}\nolimits\!\left(s\right)\neq 0 for \realpart{s}>1. Also, \mathop{\zeta\/}\nolimits\!\left(s\right)\neq 0 for \realpart{s}=1, a property first established in Hadamard (1896) and de la Vallée Poussin (1896a, b) in the proof of the prime number theorem (25.16.3). The functional equation (25.4.1) implies \mathop{\zeta\/}\nolimits\!\left(-2n\right)=0 for n=1,2,3,\dots. These are called the trivial zeros. Except for the trivial zeros, \mathop{\zeta\/}\nolimits\!\left(s\right)\neq 0 for \realpart{s}\leq 0. In the region 0<\realpart{s}<1, called the critical strip, \mathop{\zeta\/}\nolimits\!\left(s\right) has infinitely many zeros, distributed symmetrically about the real axis and about the critical line \realpart{s}=\frac{1}{2}. The Riemann hypothesis states that all nontrivial zeros lie on this line.

Calculations relating to the zeros on the critical line make use of the real-valued function

where

is chosen to make Z(t) real, and \mathop{\mathrm{ph}\/}\nolimits\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{4}+% \frac{1}{2}it\right) assumes its principal value. Because |Z(t)|=|\mathop{\zeta\/}\nolimits\!\left(\frac{1}{2}+it\right)|, Z(t) vanishes at the zeros of \mathop{\zeta\/}\nolimits\!\left(\frac{1}{2}+it\right), which can be separated by observing sign changes of Z(t). Because Z(t) changes sign infinitely often, \mathop{\zeta\/}\nolimits\!\left(\frac{1}{2}+it\right) has infinitely many zeros with t real.

§25.10(ii) Riemann–Siegel Formula

Show Annotations
Notes:
See Apostol (1976, Chapter 12), Titchmarsh (1986b, p. 89).
Keywords:
Riemann–Siegel formula, Riemann zeta function
Permalink:
http://dlmf.nist.gov/25.10.ii

Riemann developed a method for counting the total number N(T) of zeros of \mathop{\zeta\/}\nolimits\!\left(s\right) in that portion of the critical strip with 0<t<T. By comparing N(T) with the number of sign changes of Z(t) we can decide whether \mathop{\zeta\/}\nolimits\!\left(s\right) has any zeros off the line in this region. Sign changes of Z(t) are determined by multiplying (25.9.3) by \mathop{\exp\/}\nolimits\!\left(i\vartheta(t)\right) to obtain the Riemann–Siegel formula:

where R(t)=\mathop{O\/}\nolimits\!\left(t^{{-1/4}}\right) as t\to\infty.

The error term R(t) can be expressed as an asymptotic series that begins

Riemann also developed a technique for determining further terms. Calculations based on the Riemann–Siegel formula reveal that the first ten billion zeros of \mathop{\zeta\/}\nolimits\!\left(s\right) in the critical strip are on the critical line (van de Lune et al. (1986)). More than one-third of all the zeros in the critical strip lie on the critical line (Levinson (1974)).

For further information on the Riemann–Siegel expansion see Berry (1995).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available.